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Dirichlet’s principle, uniqueness of harmonic maps and extremal QC mappings. (English) Zbl 1289.30001

Stanković, Bogoljub, Two topics in mathematics. Beograd: Matematički Institut SANU (ISBN 86-80593-36-2). Zbornik Radova (Beograd) 10(18), 41-91 (2004).
From the text: This expository paper consists of various uniqueness theorems which follow, in general, from the length-area principle of Grötzsch. The structure of this paper is as follows. In Section I we give the main ideas and basic results. In the Subsections A, B, C, D and E we discuss connections between the Grötzsch principle, Teichmüller’s approach, the main inequality and Dirichlet’s principle. In the Subsections F and G we consider extremal problems for quasiconformal mappings. In particular, we give a short review of new results and solve some problems, which originally were subject of investigations of Teichmüller, Reich, Strebel and other mathematicians.
In Section II we give an outline of proofs of some properties of harmonic maps using different tools such as Dirichlet’s principle, minimizing sequences, and different versions of the Reich-Strebel inequality. We also give a proof of the well-known Beurling theorem.
In Section III, using Lemma C1 we prove the inequality of Reich and Strebel for Riemann surfaces of finite analytic type and a new version of an inequality of Reich and Strebel. We use this result to study uniqueness properties of harmonic mappings (see Section II).
Section IV is an extended version of the lecture given by the author at the 8. Romanian-Finish Seminar in Iassy in August ’99.
For the entire collection see [Zbl 1060.00006].

MSC:

30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30C62 Quasiconformal mappings in the complex plane
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30C55 General theory of univalent and multivalent functions of one complex variable
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